Definition:Empty Set

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Definition

The empty set is a set which has no elements.

It is usually denoted by some variant of a zero with a line through it, for example $\varnothing$ or $\empty$, and can always be represented as $\left\{{}\right\}$.

This website prefers $\varnothing$ for its completely unambiguous interpretation.


It is justifiable to refer to it as the empty set because there is only one empty set. That is, any two empty sets are necessarily equal, and therefore the same set.

See Empty Set Unique for a proof of this.


Axiomatic Set Theory

The concept of the empty set is axiomatised in the Axiom of Existence in Zermelo-Fraenkel set theory, where it is rigorously formalized by:

$\varnothing = \left\{{x : x \ne x}\right\}$

that is, the set of all elements which are not equal to each other.


Alternative Terms

This is sometimes called the null set, but this name is discouraged because there is another concept for null set which ought not to be confused with this.

Some sources, for example Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951), call this the vacuous set.


Notes on Symbology

The symbols $\varnothing$ and $\empty$ are properly considered as stylings of $0$ (zero), and not variants of the Greek "Phi": $\Phi, \phi, \varphi$.[1]

Some sources maintain that it is a variant on the Norwegian / Danish / Faeroese letter Ø.


Some sources use $\Box$ as the symbol for the empty set, but this is rare.


Comment

Some authors have problems with the existence (or not) of the empty set:

If $A, B$ are disjoint, then $A \cap B$ is not really defined, because it has no elements. For this reason we introduce a conventional empty set, denoted $\varnothing$, to be thought of as a 'set with no elements'. Of course this is a set only by courtesy, but it is convenient to allow $\varnothing$ the status of a set.
The best attitude towards the empty set $\varnothing$ is, perhaps, to regard it as an interesting curiosity, a convenient fiction. To say that $x \in \varnothing$ simply means that $x$ does not exist. Note that it is conveniently agreed that $\varnothing$ is a subset of every set, for elements of $\varnothing$ are supposed to possess every property.


Such a philosophical position is, to the school of anarchist Bourbakist existentialists such as Matt Westwood, a contemptibly timid attitude harking back to the mediaeval distrust of zero.


Notes

  1. ↑ This has relevance to the early days of the computing industry, when $\empty$ was frequently used to mean "zero", in order to distinguish it from $\mathrm O$ (the letter "O").
    In the same context, the letter "O" was sometimes seen rendered as $\odot$, so as to ensure its being differentiated from "zero".
    The latter has fallen out of use, but it is still common for mathematicians, when writing their mathematics by hand, to strike through their zeroes out of habit.


Sources

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